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  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#2">第2讲 函数极限</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1">1. 极限的四则运算</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2_1">2. 基本极限公式</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#3">3. 洛必达法则</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#4">4. 泰勒公式</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_1">（1）常用泰勒级数展开式</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#5">5. 无穷比介小</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#6">6.  无穷小的运算规则</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#7">7. 常用的等价无穷小</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#8">8. 函数的连续与间断</a>
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  <h1 id="数学-高等数学 1 第2讲 函数极限" class="content-subhead">数学-高等数学 1 第2讲 函数极限</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
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    <h2 id="2">第2讲 函数极限</h2>
<h3 id="1">1. 极限的四则运算</h3>
<p>前提<br />
<script type="math/tex; mode=display">
\lim f(x)=A\ 存在\\[1ex]
\lim g(x)=B\ 存在
</script>
<br />
公式<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\lim[f(x)\pm g(x)]&=\lim{f(x)}\pm\lim{g(x)}=A\pm B \\[2ex]
\lim[f(x)· g(x)]&=\lim{f(x)}·\lim{g(x)}=A· B \\[2ex]
当\ B\neq0\ 时\ \ \ \ \ \ \ \ \ \ \ \ \lim\cfrac{f(x)}{g(x)}&=\cfrac{\lim{f(x)}}{\lim{g(x)}}=\cfrac{A}{B} \\[2ex]
当\ A>0\ 时\ \ \ \ \ \ \ \lim[f(x)]^{g(x)}&=[\lim f(x)]^{\lim g(x)}=A^B
\end{split}\end{equation}
</script>
</p>
<h3 id="2_1">2. 基本极限公式</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\lim_{x\to\infty}(1+\cfrac{1}{x})^x &= e \\[1ex]
\lim_{x\to0}(1+x)^{\frac{1}{x}} &= e \\[1ex]
\lim_{x\to\infty}(1+\cfrac{a}{x})^{bx} &= e^{ab}
\end{split}\end{equation}
</script>
</p>
<blockquote class="content-quote">
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\lim_{x\to\infty}(1+\cfrac{a}{x})^{bx} &= e^{ab} \\[1ex]
y &= (1+\cfrac{a}{x})^{bx} \\[1ex]
\ln y &= bx\ln(1+\cfrac{a}{x}) \\[1ex]
&= bx\sum_{i=1}^{\infty}(-1)^{n-1}\cfrac{(\frac{a}{x})^n}{n} \\[1ex]
&= bx[\cfrac{a}{x}+o(\cfrac{a}{x})] \\[1ex]
&= ab+bxo(\cfrac{a}{x}) \\[1em]
y &= e^{ab+bxo(\frac{a}{x})} \\[1ex]
\lim_{x\to\infty}y &= e^{ab} \\[1ex]
\end{split}\end{equation}
</script>
</p>
</blockquote>
<h3 id="3">3. 洛必达法则</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\lim_{x\to·}\cfrac{f(x)}{g(x)} &= \lim_{x\to·}\cfrac{f'(x)}{g'(x)} \\[1em]
\lim_{x\to·}\cfrac{\int_{a}^{x}f(t)dt}{\int_{a}^{x}g(t)dt} &= \lim_{x\to·}\cfrac{f(x)}{g(x)} \\[1em]
\lim_{x\to·}\cfrac{\int_{\psi(x)}^{\varphi(x)}f(t)dt}{\int_{\psi(x)}^{\varphi(x)}g(t)dt} &= \lim_{x\to·}\cfrac{f[\varphi(x)]\varphi'(x)-f[\psi(x)]\psi'(x)}{g[\varphi(x)]\varphi'(x)-g[\psi(x)]\psi'(x)}
\end{split}\end{equation}
</script>
</p>
<h3 id="4">4. 泰勒公式</h3>
<video style="border: 1px solid rgba(0, 0, 0, 1);" controls="controls" width="100%" src="/post/数学-高等数学.assets/泰勒级数.mov"></video>

<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f(x) &= \sum_{n=0}^\infty \cfrac{f^{(n)}(x_0)}{n!}(x - x_0)^n \ \ \ \ \  (令x_0=0)\\[2ex]
&= \sum_{n=0}^\infty \cfrac{f^{(n)}(0)}{n!}x^n
\end{split}\end{equation}
</script>
</p>
<h4 id="1_1">（1）常用泰勒级数展开式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
   \cos x &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + ... = \sum_{n=0}^\infty(-1)^n\cfrac{x^{2n}}{(2n)!} \\[1ex]
  \cosh x &= 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + ... = \sum_{n=0}^\infty\cfrac{x^{2n}}{(2n)!} \\[1ex]
\arccos x &= \cfrac{\pi}{2} - \arcsin x \\[2em]
   \sin x &= x - \frac{x^3}{3!} + ... = \sum_{n=0}^\infty(-1)^n\cfrac{x^{2n+1}}{(2n+1)!} \\[1ex]
  \sinh x &= x + \frac{x^3}{3!} + ... = \sum_{n=0}^\infty\cfrac{x^{2n+1}}{(2n+1)!} \\[1ex]
\arcsin x &= x + \frac{x^3}{6} + o(x^3) \\[2em]
   \tan x &= x + \frac{x^3}{3} + o(x^3) \\[1ex]
\arctan x &= x - \frac{x^3}{3} + o(x^3) = \sum_{n=1}^\infty(-1)^{n-1}\cfrac{x^{2n-1}}{2n-1} \\[2em]
 \ln(1+x) &= x - \frac{x^2}{2} + \frac{x^3}{3} ... = \sum_{n=1}^\infty(-1)^{n-1}\cfrac{x^{n}}{n} &-1\lt x\le 1 \\[1ex]
 \ln(1-x) &= - \sum_{n=1}^\infty\cfrac{x^{n}}{n} &-1\lt x\le 1 \\[1ex]
\ln(1-x^2)&= -2\sum_{n=1}^\infty\cfrac{x^{2n}}{2n} &-1\lt x\le 1 \\[1ex]
\ln\cfrac{1+x}{1-x}&= 2\sum_{n=1}^\infty\cfrac{x^{2n-1}}{2n-1} &-1\lt x\le 1 \\[2em]
      e^x &= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... = \sum_{n=0}^\infty\cfrac{x^{n}}{n!} \\[2ex]
  (1+x)^a &= 1 + ax + \frac{a(a-1)}{2!}x^2 + \frac{a(a-1)(a-2)}{3!}x^3 + ... \\[2em]
\cfrac{1}{1-x} &= \sum_{n=0}^\infty x^n &|x|\lt1 \\[1ex]
\cfrac{1}{(1-x)^2} &= \sum_{n=0}^\infty (n+1)x^n &|x|\lt1 \\[1ex]
\cfrac{2}{(1-x)^3} &= \sum_{n=0}^\infty (n+2)(n+1)x^n &|x|\lt1 \\[1ex]
\end{split}\end{equation}
</script>
</p>
<blockquote class="content-quote">
<p>
<script type="math/tex; mode=display">
\arctan x = \int\cfrac{1}{1+x^2}dx
</script>
</p>
<ol>
<li>无穷级数求和函数，应灵活运用上面的泰勒级数展开式</li>
<li>在与导数相关题目中的应用：</li>
</ol>
<p>【2016年考研数一16题】设函数 <script type="math/tex">f(x)=\arctan x-\cfrac{x}{1+ax^2}</script>，且 <script type="math/tex">f'''(0)=1</script>，则 <script type="math/tex">a=</script> ____<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
f(x)&=\arctan x-\cfrac{x}{1+ax^2} \\
&=x-\cfrac{1}{3}x^3+o(x^3)-x(1-ax^2+o(x^2)) \\
&=(a-\cfrac{1}{3})x^3+o(x^3) \\[1ex]
f'''(x)&=3*2*1*(a-\cfrac{1}{3})=1 \\[1ex]
a&=\cfrac{1}{2}
\end{split}\end{equation}
</script>
</p>
<ol start="3">
<li>求泰勒公式的系数</li>
</ol>
<p>
<script type="math/tex; mode=display">
f(x) = \sum_{n=0}^\infty \cfrac{f^{(n)}(x_0)}{n!}(x - x_0)^n \\[1ex]
根据第n项的表达式\ \ \ \cfrac{f^{(n)}(x_0)}{n!}(x - x_0)^n \\[1ex]
\lim_{x\to x_0}\cfrac{f(x)}{(x - x_0)^n}=\cfrac{f^{(n)}(x_0)}{n!}=第n项系数
</script>
</p>
</blockquote>
<h3 id="5">5. 无穷比介小</h3>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th></th>
<th>表达式</th>
<th>
<script type="math/tex"> \alpha(x) </script>  是  <script type="math/tex"> \beta(x) </script>  的</th>
<th>趋向于0的速</th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>
<script type="math/tex"> \lim\cfrac{\alpha(x)}{\beta(x)} = 0 </script>
</td>
<td>高阶无穷小</td>
<td>
<script type="math/tex"> \alpha(x) 快于 \beta(x) </script>
</td>
<td>
<script type="math/tex"> \alpha(x) = o(\beta(x)) </script>
</td>
</tr>
<tr>
<td>2</td>
<td>
<script type="math/tex"> \lim\cfrac{\alpha(x)}{\beta(x)} = \infty </script>
</td>
<td>低阶无穷小</td>
<td>
<script type="math/tex"> \alpha(x) 慢于 \beta(x) </script>
</td>
<td>
<script type="math/tex"> \alpha(x) = \omega(\beta(x)) </script>
</td>
</tr>
<tr>
<td>3</td>
<td>
<script type="math/tex"> \lim\cfrac{\alpha(x)}{\beta(x)} = c \neq 0 </script>
</td>
<td>同阶无穷小</td>
<td>相近</td>
<td></td>
</tr>
<tr>
<td>4</td>
<td>
<script type="math/tex"> \lim\cfrac{\alpha(x)}{\beta(x)} = 1 </script>
</td>
<td>等阶无穷小</td>
<td>相等</td>
<td>
<script type="math/tex"> \alpha(x) ～ \beta(x) </script>
</td>
</tr>
<tr>
<td>5</td>
<td>
<script type="math/tex"> \lim\cfrac{\alpha(x)}{[\beta(x)]^k} = c \neq 0，k \gt 0 </script>
</td>
<td>
<script type="math/tex"> k </script>  阶无穷小</td>
<td></td>
<td></td>
</tr>
</tbody>
</table></div>
<h3 id="6">6.  无穷小的运算规则</h3>
<ol>
<li>有限个无穷小的和是无穷小</li>
<li>有限个无穷小的乘积是无穷小</li>
<li>有界函数与无穷小的乘积是无穷小</li>
<li>无穷小的运算</li>
</ol>
<ul>
<li>加减法：低价吸收高阶  <script type="math/tex"> o(x^2) \pm o(x^3) = o(x^2) </script>
</li>
<li>乘法：阶数累加  <script type="math/tex"> o(x^2) * o(x^3) = o(x^5) </script>
</li>
<li>非0常数相乘不影响阶数   <script type="math/tex"> o(k * x^2) = k * o(x^2) </script>
</li>
</ul>
<h3 id="7">7. 常用的等价无穷小</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
   \sin x &\sim x \\[1em]
   \tan x &\sim x \\[1em]
\arcsin x &\sim x \\[1em]
\arctan x &\sim x \\[2em]
\ln(1 + x) &\sim x \\[1ex]
\ln(x) &\sim x-1\ \ \ \   (*)\\
\log_a(1 + x) &\sim \cfrac{x}{\ln a} \\[1em]
  e^x - 1 &\sim x \\[1ex]
  a^x - 1 &\sim x\ln a \\[2em]
(1 + x)^a - 1&\sim ax \\[1em]
1 -\cos\ x &\sim \frac{1}{2}x^2
\end{split}\end{equation}
</script>
</p>
<h3 id="8">8. 函数的连续与间断</h3>
<ol>
<li>可去间断点</li>
<li>跳跃间断点</li>
<li>无穷间断点</li>
<li>震荡间断点</li>
</ol>
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